40 KiB
40 KiB
In [4]:
R.<ss, b1, b2, a1, a2, alpha1, alpha2, alpha, nu1, nu2, t, p1, p2, cd, rt> = QQ[]
In [141]:
e1, e2, e3 = [v.change_ring(R) for v in (QQ^3).gens()]
In [142]:
V2 = vector([0, 0, 3]) V1 = vector([b1, b2, 2]) V0 = vector([a1 + b1, a2 + b2, 0]) nu = vector([nu1, nu2, 1])
In [143]:
Q1 = vector([1, 0, b1]) Q2 = vector([0, 1, b2]) R1 = vector([2, 0, a1]) / alpha1 R2 = vector([0, 2, a2]) / alpha2
In [144]:
def cono3(u, v, w, vert, nu0): m = Matrix([u, v, w]).det() den = 1 for u0 in (u, v, w): den *= (u0 * vert) * ss + u0 * nu0 return m / den
In [145]:
# sb = {ss: (t - 1) / 3} sb = {}
In [146]:
vertice2 = cono3(e1, e2, e3, V2, nu) - cono3(Q1, Q2, e3, V2, nu) show(vertice2.subs(sb).factor())
<html>\(\displaystyle \nu_{2}^{-1} \cdot \nu_{1}^{-1} \cdot (3 \mathit{ss} b_{2} + b_{2} + \nu_{2})^{-1} \cdot (3 \mathit{ss} b_{1} + b_{1} + \nu_{1})^{-1} \cdot (3 \mathit{ss} b_{1} b_{2} + b_{1} b_{2} + b_{2} \nu_{1} + b_{1} \nu_{2})\)</html>
In [147]:
vertice1 = cono3(Q1, Q2, e3, V1, nu) - cono3(R1, R2, e3, V1, nu) show(vertice1.subs(sb).factor())
<html>\(\displaystyle (2 \mathit{ss} b_{2} + 2 \mathit{ss} a_{2} + a_{2} + 2 \nu_{2})^{-1} \cdot (3 \mathit{ss} b_{2} + b_{2} + \nu_{2})^{-1} \cdot (2 \mathit{ss} b_{1} + 2 \mathit{ss} a_{1} + a_{1} + 2 \nu_{1})^{-1} \cdot (3 \mathit{ss} b_{1} + b_{1} + \nu_{1})^{-1} \cdot (-16 \mathit{ss} b_{1} b_{2} + 2 \mathit{ss} b_{2} a_{1} + 2 \mathit{ss} b_{1} a_{2} + 2 \mathit{ss} a_{1} a_{2} - 4 b_{1} b_{2} + a_{1} a_{2} - 4 b_{2} \nu_{1} + 2 a_{2} \nu_{1} - 4 b_{1} \nu_{2} + 2 a_{1} \nu_{2})\)</html>
In [148]:
vertice0 = cono3(R1, R2, e3, V0, nu) show(vertice0.subs(sb).factor())
<html>\(\displaystyle 2^{2} \cdot (2 \mathit{ss} b_{2} + 2 \mathit{ss} a_{2} + a_{2} + 2 \nu_{2})^{-1} \cdot (2 \mathit{ss} b_{1} + 2 \mathit{ss} a_{1} + a_{1} + 2 \nu_{1})^{-1}\)</html>
In [149]:
Matrix([Q1, Q2]).minors(2)
Out[149]:
[1, b2, -b1]
In [150]:
mgamma1 = 1
In [151]:
Q1 * V2, Q1 * V1, Q2 * V2, Q2 * V1, Q1 * nu, Q2 * nu
Out[151]:
(3*b1, 3*b1, 3*b2, 3*b2, b1 + nu1, b2 + nu2)
In [152]:
den11 = ((Q1 * V1) * ss + Q1 * nu) den21 = ((Q2 * V1) * ss + Q2 * nu)
In [153]:
arista1 = mgamma1 * ss / (ss + 1) / den11 / den21 show(arista1.subs(sb).factor())
<html>\(\displaystyle (\mathit{ss} + 1)^{-1} \cdot \mathit{ss} \cdot (3 \mathit{ss} b_{2} + b_{2} + \nu_{2})^{-1} \cdot (3 \mathit{ss} b_{1} + b_{1} + \nu_{1})^{-1}\)</html>
In [154]:
Matrix([R1, R2]).minors(2)
Out[154]:
[4/(alpha1*alpha2), 2*a2/(alpha1*alpha2), (-2*a1)/(alpha1*alpha2)]
In [155]:
mgamma0 = 2 * cd / alpha1 / alpha2
In [156]:
R1 * V1, R1 * V0, R2 * V1, R2 * V0, R1 * nu, R2 * nu
Out[156]:
((2*b1 + 2*a1)/alpha1, (2*b1 + 2*a1)/alpha1, (2*b2 + 2*a2)/alpha2, (2*b2 + 2*a2)/alpha2, (a1 + 2*nu1)/alpha1, (a2 + 2*nu2)/alpha2)
In [157]:
den10 = ((R1 * V1) * ss + R1 * nu) den20 = ((R2 * V1) * ss + R2 * nu)
In [159]:
arista0 = mgamma0 * ss / (ss + 1) / den10 / den20 * cd show(arista0.subs(sb).factor())
<html>\(\displaystyle \left(2\right) \cdot (\mathit{ss} + 1)^{-1} \cdot \mathit{ss} \cdot \mathit{cd}^{2} \cdot (2 \mathit{ss} b_{2} + 2 \mathit{ss} a_{2} + a_{2} + 2 \nu_{2})^{-1} \cdot (2 \mathit{ss} b_{1} + 2 \mathit{ss} a_{1} + a_{1} + 2 \nu_{1})^{-1}\)</html>
In [161]:
prueba = vertice2 - arista1 + vertice1 - arista0 + vertice0 # prueba.numerator().variables() + prueba.denominator().variables() show(prueba.subs(sb).factor())
<html>\(\displaystyle \left(-1\right) \cdot \nu_{2}^{-1} \cdot \nu_{1}^{-1} \cdot (\mathit{ss} + 1)^{-1} \cdot (2 \mathit{ss} b_{2} + 2 \mathit{ss} a_{2} + a_{2} + 2 \nu_{2})^{-1} \cdot (3 \mathit{ss} b_{2} + b_{2} + \nu_{2})^{-1} \cdot (2 \mathit{ss} b_{1} + 2 \mathit{ss} a_{1} + a_{1} + 2 \nu_{1})^{-1} \cdot (3 \mathit{ss} b_{1} + b_{1} + \nu_{1})^{-1} \cdot (18 \mathit{ss}^{3} b_{1} b_{2} \nu_{1} \nu_{2} \mathit{cd}^{2} - 12 \mathit{ss}^{4} b_{1}^{2} b_{2}^{2} - 12 \mathit{ss}^{4} b_{1} b_{2}^{2} a_{1} - 12 \mathit{ss}^{4} b_{1}^{2} b_{2} a_{2} - 12 \mathit{ss}^{4} b_{1} b_{2} a_{1} a_{2} + 12 \mathit{ss}^{2} b_{1} b_{2} \nu_{1} \nu_{2} \mathit{cd}^{2} + 6 \mathit{ss}^{2} b_{2} \nu_{1}^{2} \nu_{2} \mathit{cd}^{2} + 6 \mathit{ss}^{2} b_{1} \nu_{1} \nu_{2}^{2} \mathit{cd}^{2} - 16 \mathit{ss}^{3} b_{1}^{2} b_{2}^{2} - 22 \mathit{ss}^{3} b_{1} b_{2}^{2} a_{1} - 22 \mathit{ss}^{3} b_{1}^{2} b_{2} a_{2} - 28 \mathit{ss}^{3} b_{1} b_{2} a_{1} a_{2} - 16 \mathit{ss}^{3} b_{1} b_{2}^{2} \nu_{1} - 4 \mathit{ss}^{3} b_{2}^{2} a_{1} \nu_{1} - 16 \mathit{ss}^{3} b_{1} b_{2} a_{2} \nu_{1} - 4 \mathit{ss}^{3} b_{2} a_{1} a_{2} \nu_{1} - 16 \mathit{ss}^{3} b_{1}^{2} b_{2} \nu_{2} - 16 \mathit{ss}^{3} b_{1} b_{2} a_{1} \nu_{2} - 4 \mathit{ss}^{3} b_{1}^{2} a_{2} \nu_{2} - 4 \mathit{ss}^{3} b_{1} a_{1} a_{2} \nu_{2} - 32 \mathit{ss}^{3} b_{1} b_{2} \nu_{1} \nu_{2} + 4 \mathit{ss}^{3} b_{2} a_{1} \nu_{1} \nu_{2} + 4 \mathit{ss}^{3} b_{1} a_{2} \nu_{1} \nu_{2} + 4 \mathit{ss}^{3} a_{1} a_{2} \nu_{1} \nu_{2} + 2 \mathit{ss} b_{1} b_{2} \nu_{1} \nu_{2} \mathit{cd}^{2} + 2 \mathit{ss} b_{2} \nu_{1}^{2} \nu_{2} \mathit{cd}^{2} + 2 \mathit{ss} b_{1} \nu_{1} \nu_{2}^{2} \mathit{cd}^{2} + 2 \mathit{ss} \nu_{1}^{2} \nu_{2}^{2} \mathit{cd}^{2} - 4 \mathit{ss}^{2} b_{1}^{2} b_{2}^{2} - 12 \mathit{ss}^{2} b_{1} b_{2}^{2} a_{1} - 12 \mathit{ss}^{2} b_{1}^{2} b_{2} a_{2} - 23 \mathit{ss}^{2} b_{1} b_{2} a_{1} a_{2} - 20 \mathit{ss}^{2} b_{1} b_{2}^{2} \nu_{1} - 6 \mathit{ss}^{2} b_{2}^{2} a_{1} \nu_{1} - 28 \mathit{ss}^{2} b_{1} b_{2} a_{2} \nu_{1} - 8 \mathit{ss}^{2} b_{2} a_{1} a_{2} \nu_{1} - 4 \mathit{ss}^{2} b_{2}^{2} \nu_{1}^{2} - 4 \mathit{ss}^{2} b_{2} a_{2} \nu_{1}^{2} - 20 \mathit{ss}^{2} b_{1}^{2} b_{2} \nu_{2} - 28 \mathit{ss}^{2} b_{1} b_{2} a_{1} \nu_{2} - 6 \mathit{ss}^{2} b_{1}^{2} a_{2} \nu_{2} - 8 \mathit{ss}^{2} b_{1} a_{1} a_{2} \nu_{2} - 64 \mathit{ss}^{2} b_{1} b_{2} \nu_{1} \nu_{2} - 4 \mathit{ss}^{2} b_{2} a_{1} \nu_{1} \nu_{2} - 4 \mathit{ss}^{2} b_{1} a_{2} \nu_{1} \nu_{2} + 2 \mathit{ss}^{2} a_{1} a_{2} \nu_{1} \nu_{2} - 8 \mathit{ss}^{2} b_{2} \nu_{1}^{2} \nu_{2} + 4 \mathit{ss}^{2} a_{2} \nu_{1}^{2} \nu_{2} - 4 \mathit{ss}^{2} b_{1}^{2} \nu_{2}^{2} - 4 \mathit{ss}^{2} b_{1} a_{1} \nu_{2}^{2} - 8 \mathit{ss}^{2} b_{1} \nu_{1} \nu_{2}^{2} + 4 \mathit{ss}^{2} a_{1} \nu_{1} \nu_{2}^{2} - 2 \mathit{ss} b_{1} b_{2}^{2} a_{1} - 2 \mathit{ss} b_{1}^{2} b_{2} a_{2} - 8 \mathit{ss} b_{1} b_{2} a_{1} a_{2} - 4 \mathit{ss} b_{1} b_{2}^{2} \nu_{1} - 2 \mathit{ss} b_{2}^{2} a_{1} \nu_{1} - 14 \mathit{ss} b_{1} b_{2} a_{2} \nu_{1} - 5 \mathit{ss} b_{2} a_{1} a_{2} \nu_{1} - 4 \mathit{ss} b_{2}^{2} \nu_{1}^{2} - 6 \mathit{ss} b_{2} a_{2} \nu_{1}^{2} - 4 \mathit{ss} b_{1}^{2} b_{2} \nu_{2} - 14 \mathit{ss} b_{1} b_{2} a_{1} \nu_{2} - 2 \mathit{ss} b_{1}^{2} a_{2} \nu_{2} - 5 \mathit{ss} b_{1} a_{1} a_{2} \nu_{2} - 32 \mathit{ss} b_{1} b_{2} \nu_{1} \nu_{2} - 8 \mathit{ss} b_{2} a_{1} \nu_{1} \nu_{2} - 8 \mathit{ss} b_{1} a_{2} \nu_{1} \nu_{2} - 2 \mathit{ss} a_{1} a_{2} \nu_{1} \nu_{2} - 16 \mathit{ss} b_{2} \nu_{1}^{2} \nu_{2} - 4 \mathit{ss} b_{1}^{2} \nu_{2}^{2} - 6 \mathit{ss} b_{1} a_{1} \nu_{2}^{2} - 16 \mathit{ss} b_{1} \nu_{1} \nu_{2}^{2} - b_{1} b_{2} a_{1} a_{2} - 2 b_{1} b_{2} a_{2} \nu_{1} - b_{2} a_{1} a_{2} \nu_{1} - 2 b_{2} a_{2} \nu_{1}^{2} - 2 b_{1} b_{2} a_{1} \nu_{2} - b_{1} a_{1} a_{2} \nu_{2} - 4 b_{1} b_{2} \nu_{1} \nu_{2} - 2 b_{2} a_{1} \nu_{1} \nu_{2} - 2 b_{1} a_{2} \nu_{1} \nu_{2} - a_{1} a_{2} \nu_{1} \nu_{2} - 4 b_{2} \nu_{1}^{2} \nu_{2} - 2 a_{2} \nu_{1}^{2} \nu_{2} - 2 b_{1} a_{1} \nu_{2}^{2} - 4 b_{1} \nu_{1} \nu_{2}^{2} - 2 a_{1} \nu_{1} \nu_{2}^{2} - 4 \nu_{1}^{2} \nu_{2}^{2})\)</html>
In [162]:
def compara(aa1, aa2, bb1, bb2, nnu1, nnu2): R.<x, y, z> = QQ[] F = z^3 + x^bb1 * y^bb2 * (z^2 + x^aa1 * y^aa2) zt = ZetaFunctions(F).topological_zeta(local=True, weights=(nnu1, nnu2, 1)) S = zt.parent() s = S.gen(0) aalpha1 = aa1.gcd(2) aalpha2 = aa2.gcd(2) ccd = aalpha1.gcd(aalpha2) pr = prueba(nu1=nnu1, nu2=nnu2, b1=bb1, b2=bb2, alpha1=aalpha1, alpha2=aalpha2, a1=aa1, a2=aa2, cd=ccd, ss=s) return (zt, pr)
In [163]:
zt, pr = compara(5, 7, 2, 3, 2, 3)
In [164]:
zt.factor()
Out[164]:
(1/18) * (s + 9/14)^-1 * (s + 13/20)^-1 * (s + 2/3)^-1 * (s + 1)^-1 * (s^3 + 339/140*s^2 + 97/40*s + 117/140)
In [165]:
pr.factor()
Out[165]:
(1/18) * (s + 9/14)^-1 * (s + 13/20)^-1 * (s + 2/3)^-1 * (s + 1)^-1 * (s^3 + 339/140*s^2 + 97/40*s + 117/140)
In [166]:
zt - pr
Out[166]:
0
In [167]:
zt, pr = compara(8, 12, 3, 5, 2, 5)
In [168]:
zt.factor()
Out[168]:
(1/30) * (s + 6/11)^-1 * (s + 5/9)^-1 * (s + 11/17)^-1 * (s + 1)^-1 * (s^3 + 380/187*s^2 + 1049/561*s + 10/17)
In [169]:
pr.factor()
Out[169]:
(1/30) * (s + 6/11)^-1 * (s + 5/9)^-1 * (s + 11/17)^-1 * (s + 1)^-1 * (s^3 + 380/187*s^2 + 1049/561*s + 10/17)
In [170]:
zt - pr
Out[170]:
0
In [269]:
zt, pr = compara(5, 0, 2, 0, 8, 1)
In [270]:
zt.factor()
Out[270]:
(-3/56) * (s + 1)^-1 * (s + 5/3)^-1 * (s - 35/9)
In [271]:
pr.factor()
Out[271]:
(-3/56) * (s + 1)^-1 * (s + 5/3)^-1 * (s - 35/9)
In [179]:
zt - pr
Out[179]:
0
In [201]:
zt, pr = compara(8, 0, 2, 0, 2, 1)
In [202]:
zt.factor()
Out[202]:
(-1/10) * (s + 3/5)^-1 * (s + 1)^-1 * (s - 3)
In [203]:
pr.factor()
Out[203]:
(-1/10) * (s + 3/5)^-1 * (s + 1)^-1 * (s - 3)
In [77]:
zt - pr
Out[77]:
0
In [78]:
prt = prueba.subs(sb) show(prt.factor())
<html>\(\displaystyle \left(-\frac{1}{4}\right) \cdot (t + 1)^{-1} \cdot \nu_{2}^{-1} \cdot \nu_{1}^{-1} \cdot (a_{2} t + \nu_{2})^{-1} \cdot (a_{1} t + \nu_{1})^{-1} \cdot (2 \nu_{1} \nu_{2} t \mathit{cd}^{2} - 2 a_{1} a_{2} t^{2} - \nu_{1} \nu_{2} \mathit{cd}^{2} - 2 a_{1} a_{2} t - 2 a_{2} \nu_{1} t - 2 a_{1} \nu_{2} t - 2 a_{2} \nu_{1} - 2 a_{1} \nu_{2} - 6 \nu_{1} \nu_{2})\)</html>
In [80]:
prt1 = prt * (a2 * t + nu2) * (a1 * t + nu1) * (t + 1) show(prt1.factor())
<html>\(\displaystyle \left(-\frac{1}{4}\right) \cdot \nu_{2}^{-1} \cdot \nu_{1}^{-1} \cdot (2 \nu_{1} \nu_{2} t \mathit{cd}^{2} - 2 a_{1} a_{2} t^{2} - \nu_{1} \nu_{2} \mathit{cd}^{2} - 2 a_{1} a_{2} t - 2 a_{2} \nu_{1} t - 2 a_{1} \nu_{2} t - 2 a_{2} \nu_{1} - 2 a_{1} \nu_{2} - 6 \nu_{1} \nu_{2})\)</html>
In [81]:
cf23 = (prt1 / (a2 * t + nu2) / (a1 * t + nu1))(t=-1) show(cf23.factor())
<html>\(\displaystyle \left(-\frac{3}{4}\right) \cdot (a_{2} - \nu_{2})^{-1} \cdot (-a_{1} + \nu_{1})^{-1} \cdot (\mathit{cd}^{2} + 2)\)</html>
In [82]:
cf1 = (prt1 / (t + 1) / (a2 * t + nu2))(t=-nu1 / a1) show(cf1.factor())
<html>\(\displaystyle \left(\frac{1}{4}\right) \cdot \nu_{1}^{-1} \cdot (a_{1} - \nu_{1})^{-1} \cdot a_{1} \cdot (a_{1} + 2 \nu_{1}) \cdot (-a_{2} \nu_{1} + a_{1} \nu_{2})^{-1} \cdot (\nu_{1} \mathit{cd}^{2} + 2 a_{1})\)</html>
In [83]:
cf2 = (prt1 / (t + 1) / (a1 * t + nu1))(t=-nu2 / a2) show(cf2.factor())
<html>\(\displaystyle \left(\frac{1}{4}\right) \cdot \nu_{2}^{-1} \cdot (a_{2} - \nu_{2})^{-1} \cdot a_{2} \cdot (a_{2} + 2 \nu_{2}) \cdot (a_{2} \nu_{1} - a_{1} \nu_{2})^{-1} \cdot (\nu_{2} \mathit{cd}^{2} + 2 a_{2})\)</html>
In [84]:
(cf23 / (t + 1) + cf1 / (a1 * t + nu1) + cf2 / (a2 * t + nu2) - prt)
Out[84]:
0
In [85]:
cf23(a2=0, nu2=1) / (t + 1) + cf1(a2=0, nu2=1) / (a1 * t + nu1) - prt(a2=0, nu2=1)
Out[85]:
0
In [86]:
show(cf23(a2=0, nu2=1).factor())
<html>\(\displaystyle \left(-\frac{3}{4}\right) \cdot (a_{1} - \nu_{1})^{-1} \cdot (\mathit{cd}^{2} + 2)\)</html>
In [87]:
show(cf1(a2=0, nu2=1).factor())
<html>\(\displaystyle \left(\frac{1}{4}\right) \cdot \nu_{1}^{-1} \cdot (a_{1} - \nu_{1})^{-1} \cdot (a_{1} + 2 \nu_{1}) \cdot (\nu_{1} \mathit{cd}^{2} + 2 a_{1})\)</html>
In [88]:
prta = prt(a2=rt * a1, nu2=rt * nu1) show(prta.factor())
<html>\(\displaystyle \left(-\frac{1}{4}\right) \cdot \nu_{1}^{-2} \cdot \mathit{rt}^{-1} \cdot (t + 1)^{-1} \cdot (a_{1} t + \nu_{1})^{-2} \cdot (2 \nu_{1}^{2} t \mathit{cd}^{2} - 2 a_{1}^{2} t^{2} - \nu_{1}^{2} \mathit{cd}^{2} - 2 a_{1}^{2} t - 4 a_{1} \nu_{1} t - 4 a_{1} \nu_{1} - 6 \nu_{1}^{2})\)</html>
In [89]:
cf23a = cf23(a2=rt * a1, nu2=rt * nu1) show(cf23a.factor())
<html>\(\displaystyle \left(\frac{3}{4}\right) \cdot (a_{1} - \nu_{1})^{-2} \cdot \mathit{rt}^{-1} \cdot (\mathit{cd}^{2} + 2)\)</html>
In [90]:
prtb = prta - cf23a / (t + 1)
In [91]:
cf1a = (prtb * (a1 * t + nu1)^2)(t=-nu1 / a1) show(cf1a.factor()) prtc = prtb - cf1a / (a1 * t + nu1)^2 cf2a = prtc * (a1 * t + nu1) show(cf2a.factor())
<html>\(\displaystyle \left(\frac{1}{4}\right) \cdot \mathit{rt}^{-1} \cdot \nu_{1}^{-1} \cdot (a_{1} - \nu_{1})^{-1} \cdot (a_{1} + 2 \nu_{1}) \cdot (\nu_{1} \mathit{cd}^{2} + 2 a_{1})\)</html>
<html>\(\displaystyle \left(-\frac{1}{4}\right) \cdot \nu_{1}^{-2} \cdot (a_{1} - \nu_{1})^{-2} \cdot \mathit{rt}^{-1} \cdot a_{1} \cdot (3 \nu_{1}^{2} \mathit{cd}^{2} - 2 a_{1}^{2} + 4 a_{1} \nu_{1} + 4 \nu_{1}^{2})\)</html>
In [93]:
(cf23a / (t + 1) + cf1a / (a1 * t + nu1)^2 + cf2a / (a1 * t + nu1) - prta)
Out[93]:
0
In [ ]: